[Proof of Lemma REF ]
In view of Corollary 4.36 in [1]}, \(\tilde{K}_\alpha \in C^{2m}([0,1]\times [0,1])\)  implies that \(f\in C^m[0,1]\)  for any \(f \in \tilde{\mathbb {H}}_\alpha \) . This is also true for \(f \in \mathbb {H}_\alpha \)  since \(\mathbb {H}_\alpha \)  and \(\tilde{\mathbb {H}}_\alpha \)  contain the same functions.
